Now, to prove Theorem  REF(a) we apply [1]} to updateUpdate (REF ) runs over the unbounded set \(\mathbb {R}_+^d\) , unlike the compact set in [1]}. However, that result still holds here and suffices as we set \(\lambda \!=\!0\)  to get \(q_{0:T-1}(0)=0\) ; see discussion and Lemma REF  in Appendix. (REF ) with functions \(\lbrace -\lambda g_t(z_t)\rbrace \)  and gradient predictions \(\lbrace -\tilde{g}_{t+1}(\tilde{x}_{t+1})\rbrace \) , to get:
\(&-\sum _{t=1}^T \lambda _t^\top g_t(z_t)+ \sum _{t=1}^T \lambda ^\top g_t(z_t) \le \\&q_{0:T-1}(\lambda )+ \sum _{t=1}^T \Vert g_t(z_t)-\tilde{g}_t(\tilde{x}_t)\Vert _{(t-1),\star }^2 \stackrel{(a)}{=} \\&q_{0:T-1}(\lambda )+ \sum _{t=1}^T \frac{\Vert g_t(z_t)-\tilde{g}_t(\tilde{x}_t)\Vert ^2}{\phi _{0:t-1}},\,\,\,\forall \lambda \in \mathbb {R}_+^d, \) 
